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Define $X=\lbrace (0,y) \in \mathbb{R}^2|y \in \mathbb{R} \rbrace$ and $Y=\lbrace (x,y) \in \mathbb{R}^2|x>0, y=$sin$(\frac{1}{x})\rbrace$. Prove that $X \cup Y$ is connected but not path connected.

I know the definitions of connected and path connected spaces, but I don't know how to start with this exercise. Can anyone help me out?

Martin Sleziak
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  • The idea is that while any open set containing $X$ in the plane also contains parts of $Y$, thus making the two parts connected, there is no continuous path from any point on $Y$ to any point on $X$. – Arthur Dec 10 '16 at 17:31
  • This is a slightly altered version of a famous example called the topologist's sine curve, which you can find details on easily with a google search – TY Mathers Dec 10 '16 at 17:44

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